Prev: Question about Hugh Woodin's original proof of projective determinacy.
Next: Dawkins on Turing
From: byron on 29 May 2010 07:14
colin leslie dean points out The fundamental problem with Godels
theorems
are he creates an imprdeicative statement and the theorems apply to
themselves ie are impredicative- thus leading to paradox

ieie this is godels impredicative statement used in his first theorem
http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Meaning_of_the_first_incompleteness_theorem
Quote:
the corresponding Gödel sentence G asserts: “G cannot be proved to be
true within the theory T


as was pointed out many years ago if you use or create impredicative
statements then what you will get is paradox
and as dean has shown
that is what happens with godels theorems

philosophers such as russell
mathematicians such as poincare
have outlawed these statements from mathematics as they lead to
paradox in maths

and they lead to paradox in godels theorems
why
because if godels theorems are true
then they apply to godels theorems as well

if godels first theorem is true then it applies to it self



Quote:
it is shown by colin leslie dean that Godels first theorem ends in
paradox

it is said godel PROVED
"there are mathematical true statements which cant be proven"
in other words
truth does not equate with proof.

if that theorem is true
then his theorem is false

PROOF
for if the theorem is true
then truth does equate with proof- as he has given proof of a true
statement
but his theorem says
truth does not equate with proof.
thus a paradox


if godels second theorem is true
then it applies to it self
Quote:

http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Proof_sketch_for_the_second_theorem

The following rephrasing of the second theorem is even more unsettling
to the foundations of mathematics:

If an axiomatic system can be proven to be consistent and complete
from within itself, then it is inconsistent.



now this theorem ends in self-contradiction

http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf

But here is a contradiction Godel must prove that a system cannot be
proven to be consistent based upon the premise that the logic he uses
must be consistent . If the logic he uses is not consistent then he
cannot make a proof that is consistent. So he must assume that his
logic is consistent so he can make a proof of the impossibility of
proving a system to be consistent. But if his proof is true then he
has proved that the logic he uses to make the proof must be
consistent, but his proof proves that this cannot be done


as was pointed out many years ago if you use or create impredicative
statements then what you will get is paradox
and as dean has shown
that is what happens with godels theorems
From: Charlie-Boo on 29 May 2010 22:26
On May 29, 7:14 am, byron <spermato...(a)yahoo.com> wrote:

> http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf

GODEL’S INCOMPLETENESS THEOREM. ENDS IN ABSURDITY OR MEANINGLESSNESS
GODEL IS A COMPLETE FAILURE AS HE ENDS IN UTTER MEANINGLESSNESS
CASE STUDY IN THE MEANINGLESSNESS OF ALL VIEWS

Does that include your view? It seems that your paper suffers from
the same self-referential paradox that doomed Godel's proof. I
believe that you have refuted the validity of Mathematics - a great
generalization of Godel's results. And so you reaffirm Godel after
all (not surprisingly).

Then how about this proof [I think this is new - for even me I mean]:
Off the top of my head,

A = "This is not provable."
B = "This is refutable."
C = "This is not true."

Each of these asserts that it has a rather negative property if we are
to believe in it. (It comes as a surprise!) So what is the truth
value of each of these?

A is TRUE
B is FALSE
C is NEITHER

All subsets of {TRUE,FALSE}! [This suggests a shorter proof.]

So A'=B ^ A'=C ^ B'=C

A'=B : NOT PROVABLE is not REFUTABLE. Rosser's 1936 Extension to
Godel 1931
A'=C : PROVABLE is not TRUE. Godel's 1st Incompleteness Theorem (1931)
B'=C : REFUTABLE is not NOT TRUE. Smullyan's Dual Form Theorem
(1960's)

Example detailed proof of 1 of the above 3:

NOT PROVABLE is not REFUTABLE

Let PR be provable and DIS be refutable. If CONSISTENT and COMPLETE
then DIS => ~PR and ~PR => DIS, so DIS <=> ~PR but the above shows
that REFUTABLE is not NOT PROVABLE, so if CONSISTENT then not
COMPLETE, which is Rosser's 1936 result.

Vote plz:

A. Not interested in the subject.
B. Doesn't make much sense to me.
C. Makes sense - have no idea if it is well-known or not.
D. Makes sense - probably well-known.
E. Makes sense - but has no value.
F. Cool! (Really!)
G. Interesting . . . I have a question . . .
H. Given the context of all this I believe it is insipid.

C-B

1. Proves 3 huge theorems.
2. The proof is tiny.
3. It uses the familiar silly sentences and more.
4. In particular, the Liar is just a simple statement - as we should
expect it to be given its own simplicity.
5. The resolution of the Liar fits in exactly with Godel and Rosser's
results. It is the other subset of {TRUE,FALSE}. Godel is {TRUE} and
Rosser is {FALSE} and Liar is the other subset {}.
 | 
Pages: 1
Prev: Question about Hugh Woodin's original proof of projective determinacy.
Next: Dawkins on Turing